Monday Math 147

Given two non-zero complex numbers z1 and z2 such that , show that the arguments of z1 and z2 differ by π/2.

Recall that for any complex number z, , thus

Now, recall that for any complex number z, , and since , we see that, and so,
and we proved last week that z1 and z2 have arguments differing by π/2 (and thus give perpendicular vectors) if and only if .

Consider the geometric interpretation. Given a parallelogram with sides given by the vectors corresponding to z1 and z2, then z1+z2 and z1–z2 correspond to the diagonals; thus, the above is equivalent to stating that if the diagonals of a parallelogram are of equal length, then the parallelogram is a rectangle.

One Response to “Monday Math 147”

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